AoPS Academy
be the circumcircle of a scalene triangle 
. Let 
be a circle internally tangent to in 
. Tangents from 
touch in 
and 
, such that lies in the interior of 
. Similarly, tangents from 
touch in 
and 
, such that lies in the interior of .
and 
concur on the bisector of 
.
days working 
hours per day to complete a certain task whereas Seema would take 
days working 
hours per day to complete the same task. They decide to work together to complete this task. Seema agrees to work for double the number of hours per day as Renu, while Renu agrees to work for double the number of days as Seema. If Renu works 
hours per day, then the number of days Seema will work, is
. The center 
of the semi-circle is the midpoint of segment 
. We have 
and 
..
the orthocenter of . Construct line 
through perpendicular to 
. Construct the line through perpendicular to 
and let it intersect at 
. Let the line though perpendicular to 
meet 
respectively at 
. These are the required points. the orthocenter of 
and using that we finally constructed 
,the miquel point of 
is the infinity point of the newton line so we can construct 
then we draw a line parallel to through ; it hits 
at another point say 
let the line 
cuts 
at point that is exactly 
. Finally we draw a parallel to to construct 
.
and two vectors 
, construct a line , with direction 
, such that Newton line of the quadrilateral formed by the four lines 
has direction 
. form a triangle . Let 
be the parabola inscribed in its medial triangle and whose axis has the direction of the vector 
. Let be the conjugate diameter of the direction of the vector . and intersects at a point 
. The tangent at to is Newton's line of the quadrilateral 
, which has the direction of the vector .
be two lines .
s.t. 
and the newton line of 
is parallel to 